# Designing representations of trigonometry instruction to study the rationality of community college teaching

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## Abstract

We describe the process followed to design representations of mathematics teaching in a community college. The end product sought are animated videos to be used in investigating the practical rationality that community college instructors use to justify norms of the didactical contract or possible departures from those norms. We have chosen to work within the trigonometry course, in the context of an instructional situation, “finding the values of trigonometric functions,” and specifically on a case of this situation that occurs as instructors and students are working on examples on the board. We describe the design of the material needed to produce the animations: (1) identifying an instructional situation, (2) identifying norms of the contract that are key in that situation, (3) selecting or creating a scenario that illustrates those norms, (4) proposing alternative scenarios that instantiate breaches of those norms, and (5) anticipating justifications or rebuttals for the breaches that could be found in instructors’ reactions. We illustrate the interplay of contextual and theoretical elements as we make decisions and state hypothesis about the situation that will be prototyped.

## Keywords

Representation of teaching Community colleges Trigonometry Didactical contract Classroom interaction Teacher thinking Teaching practice## Notes

### Acknowledgments

The research reported in this article is supported by the National Science Foundation grants DRL-0745474 to the first author and ESI-0353285 to the second author. Opinions expressed here are the sole responsibility of the authors and do not reflect the views of the foundation.

## References

- Bailey, T. R., & Morest, V. S. (2006).
*Defending the community college equity agenda*. Baltimore: Johns Hopkins University Press.Google Scholar - Balacheff, N., & Gaudin, N. (2010). Modeling students’ conceptions: The case of function.
*Research in Collegiate Mathematics Education,**16*, 183–211.Google Scholar - Barnett, R. A., Ziegler, M. R., & Byleen, K. E. (2006).
*Analytic trigonometry with applications*(9th ed ed.). Hoboken, NJ: Wiley.Google Scholar - Blair, R. (Ed.). (2006).
*Beyond crossroads: Implementing mathematics standards in the first two years of college*. Memphis, TN: American Mathematical Association of Two Year Colleges.Google Scholar - Brousseau, G., & Otte, M. (1991). The fragility of knowledge. In A. Bishop, S. Mellin-Olsen, & J. van Dormolen (Eds.),
*Mathematical knowledge: Its growth through teaching*(pp. 13–36). Dordrecht, The Netherlands: Kluwer.Google Scholar - Cobb, P. (2000). Conducting teaching experiments in collaboration with teachers. In A. Kelly & R. Lesh (Eds.),
*Handbook of research design in mathematics and science education*(pp. 307–334). Mahwah, NJ: Erlbaum.Google Scholar - Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research.
*Educational Researcher,**32*(1), 9–13.CrossRefGoogle Scholar - Cox, R. D. (2009).
*The college fear factor: How students and professors misunderstand one another*. Boston, MA: Harvard University Press.Google Scholar - Garfinkel, H., & Sacks, H. (1970). On formal structures of practical action. In J. McKinney & E. Tiryakian (Eds.),
*Theoretical sociology: Perspectives and development*. New York: Appleton-Century-Crofts.Google Scholar - Grubb, N. W., & Associates. (1999).
*Honored but invisible: An inside look at teaching in community colleges*. New York: Routledge.Google Scholar - Herbst, P. (2006). Engaging students in proving: A double bind on the teacher.
*Journal for Research in Mathematics Education,**33*, 176–203.CrossRefGoogle Scholar - Herbst, P. (April, 2010).
*What practical rationality is*. Paper presented at the research pre-session of the annual meeting of the NCTM, San Diego, CA.Google Scholar - Herbst, P., & Balacheff, N. (2009). Proving and knowing in public: What counts as proof in a classroom. In D. Stylianou, M. Blanton, & E. J. Knuth (Eds.),
*Teaching and learning proofs across the grades*(pp. 40–64). New York: Routledge.Google Scholar - Herbst, P., & Chazan, D. (2003a). Exploring the practical rationality of mathematics teaching through conversations about videotaped episodes: The case of engaging students in proving.
*For the Learning of Mathematics,**23*(1), 2–14.Google Scholar - Herbst, P., & Chazan, D. (2003b).
*ThEMaT: Thought Experiments in Mathematics Teaching. Proposal to the National Science Foundation, Education and Human Resources Directorate, Division of Elementary, Secondary, and Informal Education, Teachers’ Professional Continuum Program*. Ann Arbor: University of Michigan.Google Scholar - Herbst, P., & Chazan, D. (2006).
*Producing a viable story of geometry instruction: What kind of representation calls forth teachers’ practical rationality?*Paper presented at the 28th annual meeting of the North American chapter of the international group of the psychology of mathematics education, Mérida, Mexico.Google Scholar - Herbst, P., Chen, C., Weiss, M., & González, G. (2009). “Doing proofs” in geometry classrooms. In D. Stylianou, M. Blanton, & E. J. Knuth (Eds.),
*Teaching and learning of proof across the grades*(pp. 250–268). New York: Routledge.Google Scholar - Herbst, P., & Miyakawa, T. (2008). When, how, and why prove theorems: A methodology to study the perspective of geometry teachers.
*ZDM The International Journal on Mathematics Education,**30*, 469–486.CrossRefGoogle Scholar - Herbst, P., Nachlieli, T., & Chazan, D. (2010). Studying the practical rationality of mathematics teaching: What goes into “installing” a theorem in geometry?
*Cognition and Instruction*(in press).Google Scholar - Keim, M. C., & Biletzky, P. E. (1999). Teaching methods used by part-time community college faculty.
*Community College Journal of Research and Practice,**23*(8), 727–737.CrossRefGoogle Scholar - Kelly, A. (2004). Design research in education: Yes, but is it methodological?
*Journal of the Learning Sciences,**13*, 115–128.CrossRefGoogle Scholar - Lutzer, D. J., Rodi, S. B., Kirkman, E. E., & Maxwell, J. W. (2007).
*Statistical abstract of undergraduate programs in the mathematical sciences in the United States: Fall 2005 CBMS Survey*. Washington, DC: American Mathematical Society.Google Scholar - Mesa, V. (2004). Characterizing practices associated with functions in middle school textbooks: An empirical approach.
*Educational Studies in Mathematics,**56*, 255–286.CrossRefGoogle Scholar - Mesa, V. (2010a). Similarities and differences in classroom interaction between remedial and pre-STEM college mathematics classrooms in a community college.
*Journal of Excellence in College Teaching*(accepted).Google Scholar - Mesa, V. (2010b).
*Achievement goal orientation of community college mathematics students and the misalignment of instructors’ perceptions*. Ann Arbor, MI: University of Michigan (manuscript under review).Google Scholar - Mesa, V. (2010c). Strategies for controlling the work in mathematics textbooks for introductory calculus.
*Research in Collegiate Mathematics Education*,*16*, 235–265.Google Scholar - Mesa, V., & John, G. (2009).
*Textbook analysis*. Ann Arbor: University of Michigan (unpublished manuscript).Google Scholar - Michener, E. R. (1978).
*Understanding understanding mathematics*. Cambridge, MA: Massachusetts Institute of Technology.Google Scholar - Midgley, C., Maehr, M. L., Hruda, L. Z., Anderman, E., & Others. (2000).
*Manual for the patterns of adaptive learning scales*. Ann Arbor, MI: University of Michigan.Google Scholar - Schuetz, P. (2002). Instructional practices of part-time and full-time faculty.
*New Directions for Community Colleges,**118*, 39–46.CrossRefGoogle Scholar - Seidman, E. (1985).
*In the words of the faculty: Perspectives on improving teaching and educational quality in community colleges*. San Francisco, CA: Jossey-Bass.Google Scholar - Suh, H., Mesa, V., Blake, T., & Whittemore, T. (October, 2010).
*An analysis of examples in college algebra textbooks: Opportunities for student learning*. Paper presented at the annual meeting of the Michigan chapter of the American mathematical association of two-year colleges, Muskegon, MI.Google Scholar - Watson, A., & Mason, J. (2005).
*Mathematics as a constructive activity*. Mahwah, NJ: Erlbaum.Google Scholar - Watson, A., & Shipman, S. (2008). Using learner generated examples to introduce new concepts.
*Educational Studies in Mathematics,**69*(2), 97–109.CrossRefGoogle Scholar - Wheeler, D. L., & Montgomery, D. (2009). Community college students’ views on learning mathematics in terms of their epistemological beliefs: A Q method study.
*Educational Studies in Mathematics,**72*, 289–306.CrossRefGoogle Scholar - Zaslavski, O. (2005). Seizing the opportunity to create uncertainty in learning mathematics.
*Educational Studies in Mathematics,**60*, 297–321.CrossRefGoogle Scholar - Zhu, X., & Simon, H. (1987). Learning mathematics from examples and by doing.
*Cognition and Instruction,**44*, 137–166.CrossRefGoogle Scholar